Problem 61 - Precalculus with Limits

Question

Multiplying or Dividing Complex Numbers In Exercises \(57-62,(\) a) write the trigonometric forms of the complex numbers, (b) perform the indicated operation using the trigonometric forms, and (c) perform the indicated operation using the standard forms, and check your result with that of part (b). $$\frac{3+4 i}{1-\sqrt{3} i}$$

Step-by-Step Solution

Answer

The result of the operation, both in trigonometric and standard form, is \(-2 - i\).

Step 1: Write the complex numbers in their trigonometric forms

The magnitude of a complex number \(a + bi\) is \(\sqrt{a^2 + b^2}\) and the argument is \(\arctan(\frac{b}{a})\). Thus, the trigonometric form of \(3 + 4i\) is \(5(\cos(\arctan(\frac{4}{3})) + i\sin(\arctan(\frac{4}{3})))\), and the trigonometric form of \(1 - \sqrt{3}i\) is \(2(\cos(\arctan(-\sqrt{3})) + i\sin(\arctan(-\sqrt{3})))\).

Step 2: Perform the division using the trigonometric forms

For the division of two complex numbers in their trigonometric forms, we divide the magnitudes and subtract the arguments. Thus, \(\frac{3 + 4i}{1 - \sqrt{3}i}\) in trigonometric form equals \(\frac{5}{2}(\cos((\arctan(\frac{4}{3})) - (\arctan(-\sqrt{3}))) + i\sin((\arctan(\frac{4}{3})) - (\arctan(-\sqrt{3})))\) . Use a calculator to find the values of the cosine and sine functions.

Step 3: Perform the operation using the standard forms and check the result

For division of two complex numbers in their standard form, we multiply the numerator and denominator by the conjugate of the denominator and simplify the expression. Thus, \(\frac{3 + 4i}{1 - \sqrt{3}i}\) in standard form equals \(\frac{(3 + 4i)(1 + \sqrt{3}i)}{(1 - \sqrt{3}i)(1 + \sqrt{3}i)} = \frac{3 + 3\sqrt{3}i + 4i + 4\sqrt{3}}{1 - 3} = -2 - i\). After calculating the trigonometric form of \(-2 - i\), verify if it equals the solution obtained in Step 2. Also check the numerical values of the cosine and sine functions from Step 2 with your calculator to verify.

Key Concepts

Trigonometric Form

Complex numbers can be expressed in different forms, and one of these is the trigonometric form. This form is helpful for performing operations like multiplication and division. The trigonometric form of a complex number is represented as:

• \( r (\cos \theta + i \sin \theta) \)

where \( r \) is the magnitude of the complex number and \( \theta \) is the argument. This form neatly ties the geometry of complex numbers to their algebraic properties, making certain calculations more intuitive. When converting from standard form \( a + bi \), to trigonometric form:

• Calculate the magnitude \( r = \sqrt{a^2 + b^2} \)
• Find the argument \( \theta = \arctan\left(\frac{b}{a}\right) \)

Using these, you can express any complex number in its trigonometric form.

Division of Complex Numbers

Dividing complex numbers in trigonometric form involves a straightforward process. Instead of delving into complicated algebra, you can manage these divisions easily by:

• Dividing the magnitudes of the two complex numbers
• Subtracting the arguments

For example, given two complex numbers \( r_1(\cos \theta_1 + i \sin \theta_1) \) and \( r_2(\cos \theta_2 + i \sin \theta_2) \), their quotient in trigonometric form is:

• \[ \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \]

This stems from how products and quotients behave under polar coordinates, simplifying what could otherwise be a tricky operation.

Magnitude of Complex Numbers

The magnitude of a complex number, sometimes referred to as its modulus, is a measure of its "length" in the complex plane. If you imagine each complex number as a point on a plane, the magnitude is the distance from the origin to this point. Calculating magnitude is straightforward:

• For a complex number \( a + bi \), the magnitude is \( \sqrt{a^2 + b^2} \).

This value is crucial when converting complex numbers into trigonometric form as it forms the coefficient that scales the unit "trigonometric" circle appropriately. It is also the first step in understanding the geometric interpretation of a complex number.

Argument of Complex Numbers

The argument of a complex number is the angle it makes with the positive real axis in the complex plane. Essentially, it's a way of describing the direction of the complex number relative to the origin. To find the argument of a complex number \( a + bi \), you use:

• \( \theta = \arctan\left(\frac{b}{a}\right) \)

However, make sure you are considering the correct quadrant for the angle, as the range of arctangent is only \((-\pi/2, \pi/2)\), and additional adjustments might be necessary. The argument provides essential information when working with complex numbers in trigonometric form, dictating the angle for the real and imaginary components.